# Quadratic Spline Fitting for Robust Measurement of Thoracic Kyphosis Using Key Vertebral Landmarks

## TL;DR

This paper introduces a new method for measuring thoracic kyphosis using quadratic splines, which is more accurate and reliable than traditional methods.

## Contribution

A novel quadratic spline-based method for measuring thoracic kyphosis is proposed and validated.

## Key findings

- The spline method showed lower MAE and RMSE compared to the circle fitting method.
- Both methods demonstrated strong correlation with Cobb angles and excellent classification performance.
- The spline method is particularly robust in cases of severe spinal deformity.

## Abstract

Objective: The purpose of this study is to present a kyphosis measurement method based on quadratic spline fitting through three key vertebral landmarks: T12, T8 and T4. This approach aims to capture thoracic spine curvature more continuously and accurately than traditional methods such as the Cobb angle and circle fitting. Methods: A dataset of 560 lateral thoracic spine radiographs was retrospectively analyzed, including cases of postural kyphosis, Scheuermann’s disease, osteoporosis-induced kyphosis and ankylosing spondylitis. Two trained raters independently performed three repeated landmark annotations per image. The kyphosis angle was computed using two methods: (1) a quadratic spline fitted through the three landmarks, with the angle derived from tangent vectors at T12 and T4; and (2) a least-squares circle fit with the angle subtended between T12 and T4. Agreement with reference Cobb angles was evaluated using Pearson correlation, MAE, RMSE, ROC analysis and Bland–Altman plots. Reliability was assessed using intraclass correlation coefficients (ICC). Results: Both methods showed excellent intra- and inter-rater reliability (ICC ≥ 0.967). The spline method achieved lower MAE (5.81°), lower RMSE (8.94°) and smaller bias compared to the circle method. Both methods showed strong correlation with Cobb angles (r ≥ 0.851) and excellent classification performance (AUC > 0.950). Conclusions: Spline-based kyphosis measurement is accurate, reliable and particularly robust in cases with severe spinal deformity. Significance: This method supports automated, reproducible kyphosis assessment and may enhance clinical evaluation of spinal curvature using artificial intelligence-driven image analysis.